Orthogonal sequence

However, since the expanded system is indefinite the minimization argument of the CG-method becomes ineffective. The CG-method is thus modified by replacing the orthogonal sequence of residuals by two mutually orthogonal sequences. In addition, the conjugacy constraint of the search directions is replaced by a corresponding conjugacy constraint of the two mutual search directions. 

Theorem 3.2 Each polynomial in an orthogonal sequence has all its roots real, distinct, and strictly inside the integration interval (which is also called the orthogonality interval). The roots of each polynomial lie strictly between the roots of the next higher-order polynomial in the sequence (interlacing roots). 

In this study, the appearance and evolution sequence of planar slip bands, in addition to a dislocation cell structure with increasing e,, is identical to that observed in quasi-static studies of the effects of stress path changes on dislocation substructure development [27]. The substructure evolution in copper deformed quasi-statically is known to be influenced by changes in stress path [27]. Deforming a sample in tension at 90° orthogonal to the... 

An obvious outcome of the Hantzsch synthesis is the symmetrical nature of the dihydropyridines produced. A double protection strategy has been developed to address this issue. The protected chalcone 103 was reacted with an orthogonally protected ketoester to generate dihydropyridine 104. Selective deprotection of the ester at C3 could be accomplished and the resultant acid coupled with the appropriate amine. Iteration of this sequence with the C5 ester substituent ultimately gave rise to the unsymmetrical 1,4-dihydropyridine 105. 

A Hessenberg form H (the same form but not the same matrix) can also be obtained by a sequence of orthogonal transformations, either by plane rotations (the method of Givens), each rotation annihilating an individual element, or by using unitary hermitians, I — 2wiwf, wfwt = 1 (the method of Householder), each of which annihilates ill possible elements in a column. Thus, at the first step, if A = A, and... 

Product operator A set of orthogonal operators that can be connected together by a product. The product operator approach can be used to analyze pulse sequences by applying simple mathematical rules to weakly coupled spin systems. 

The product of two orthogonal matrices is an orthogonal matrix as shown by the following sequence... 

Fig. 1.9 Schematic plot of a basic three-dimensional imaging pulse sequence with frequency encoding along one axis (read), and phase encoding along the two remaining orthogonal directions. The choice of directions is arbitrary, as is the position of the phase gradients within the sequence.

Figure 5.5.8 shows the volume selective spectroscopy pulse sequence used. This pulse sequence combines elements of NMR spectroscopy and MRI pulse sequences three slice selective rf pulses are applied in three orthogonal directions to obtain 1H spectra from pre-determined local volumes within the sample [30]. In this particular application, spectra were recorded from local volumes of dimension 1.5 mm x 1.5 mm x 0.5 mm within the fixed bed the data acquisition time for each spectrum was 3 min. Figure 5.5.9(a) shows the local volumes selected within slice 3 of the bed, as identified in Figure 5.5.7. In Figure 5.5.9(b) the 1H spectra recorded from within these volumes are shown data are presented only for the range of... 

Fig. 5.5.14 Schematic diagram showing how the double-phase encoded DEPT sequence achieves both spatial and spectral resolution within the reactor, (a) A spin-echo ]H 2D image taken through the column overlayed with a grid showing the spatial location within the column of the two orthogonal phase encoded planes (z and x) used in the modified DEPT sequence. The resulting data set is a zx image with a projection along y. In-plane spatial resol-ution is 156 [Am (z) x 141 [xm (x) for a 3-mm slice thickness. The center of each volume from which the data have been acquired is identified by the intersection of the white lines. The arrow indicates the direction of flow.

Duct flows can be converted into efficient mixing flows (i.e., flows with an exponential stretch of material lines with time) by time- modulation or by spatial changes along the duct axis. One example of the spatially periodic class is the partitioned-pipe mixer (PPM). This flow consists of a pipe partitioned with a sequence of orthogonally placed rectangular plates (Fig. 5a). The cross-sectional motion is induced through rotation of the pipe with respect to the assembly of plates, whereas the axial flow is caused by... 

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